The annealed free energy of the output distribution from a rate-R random code decomposes into bulk and sparse branches whose competition yields an explicit phase diagram in the (β, R) plane meeting at the mutual information point.
An inequality on guessing and its application to sequential decoding,
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A Statistical-Physics Refinement of Soft Covering
The annealed free energy of the output distribution from a rate-R random code decomposes into bulk and sparse branches whose competition yields an explicit phase diagram in the (β, R) plane meeting at the mutual information point.